qml.qchem.gaussian_overlap¶
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gaussian_overlap(la, lb, ra, rb, alpha, beta)[source]¶ Compute overlap integral for two primitive Gaussian functions.
The overlap integral between two Gaussian functions denoted by \(a\) and \(b\) can be computed as [Helgaker (1995) p803]:
\[S_{ab} = E^{ij} E^{kl} E^{mn} \left (\frac{\pi}{p} \right )^{3/2},\]where \(E\) is a coefficient that can be computed recursively, \(i-n\) are the angular momentum quantum numbers corresponding to different Cartesian components and \(p\) is computed from the exponents of the two Gaussian functions as \(p = \alpha + \beta\).
- Parameters
la (integer) – angular momentum for the first Gaussian function
lb (integer) – angular momentum for the second Gaussian function
ra (float) – position vector of the first Gaussian function
rb (float) – position vector of the second Gaussian function
alpha (array[float]) – exponent of the first Gaussian function
beta (array[float]) – exponent of the second Gaussian function
- Returns
overlap integral between primitive Gaussian functions
- Return type
array[float]
Example
>>> la, lb = (0, 0, 0), (0, 0, 0) >>> ra, rb = np.array([0., 0., 0.]), np.array([0., 0., 0.]) >>> alpha = np.array([np.pi/2]) >>> beta = np.array([np.pi/2]) >>> o = gaussian_overlap(la, lb, ra, rb, alpha, beta) >>> o array([1.])